On Congruent Numbers

With respect to some classification of Pythagorean triples, if anumber 𝑘 is congruent then it can easily be proven. This expandsthe quest to resolve the congruent number problem. A proposi-tion is put forward on rational sides forming a congruent number.

New Semi-Prime Factorization and Application in Large RSA Key Attacks

Semi-prime factorization is an increasingly important number theoretic problem, since it is computationally intractable. Further, this property has been applied in public-key cryptography, such as the Rivest–Shamir–Adleman (RSA) encryption systems for secure digital communications. Hence, alternate approaches to solve the semi-prime factorization problem are proposed. Recently, Pythagorean tuples to factor semi-primes have been explored to consider Fermat’s Christmas theorem, with the two squares having opposite parity. This paper is motivated by the property that the integer separating these two squares being odd reduces the search for semi-prime factorization by half. In this paper, we prove that if a Pythagorean quadruple is known and one of its squares represents a Pythagorean triple, then the semi-prime is factorized. The problem of semi-prime factorization is reduced to the problem of finding only one such sum of three squares to factorize a semi-prime. We modify the Lebesgue identity as the sum of four squares to obtain four sums of three squares. These are then expressed as four Pythagorean quadruples. The Brahmagupta–Fibonacci identity reduces these four Pythagorean quadruples to two Pythagorean triples. The greatest common divisors of the sides contained therein are the factors of the semi-prime. We then prove that to factor a semi-prime, it is sufficient that only one of these Pythagorean quadruples be known. We provide the algorithm of our proposed semi-prime factorization method, highlighting its complexity and comparative advantage of the solution space with Fermat’s method. Our algorithm has the advantage when the factors of a semi-prime are congruent to 1 modulus 4. Illustrations of our method for real-world applications, such as factorization of the 768-bit number RSA-768, are established. Further, the computational viabilities, despite the mathematical constraints and the unexplored properties, are suggested as opportunities for future research.

Direct proof through an equilateral hyperbola to find the infinite Pythagorean triples and a direct proof to find the sum of all odd numbers through a telescopic series

In this article a prove new properties like a connection with Pythagorean triples and a hiperbolic equation. I prove a direct proof about a serie, a telescopic series about the even sum and odd sum. I had an article that use this properties but this results is mine. I discovered this theorem and prove in this article.

Triangle of Triangular Numbers

Among several interesting number triangles that exist in mathematics, Pascal’s triangle is one of the best triangle possessing rich mathematical properties. In this paper, I will introduce a number triangle containing triangular numbers arranged in particular fashion. Using this number triangle, I had proved five interesting theorems which help us to generate Pythagorean triples as well as establish bijection between whole numbers and set of all integers.

PASCAL TRIANGLE AND PYTHAGOREAN TRIPLES

The concept of Pascal’s triangle has fascinated mathematicians for several centuries. Similarly, the idea of Pythagorean triples prevailing for more than two millennia continue to surprise even today with its abundant properties and generalizations. In this paper, I have demonstrated ways through four theorems to determine Pythagorean triples using entries from Pascal’s triangle.

On Pythagorean Triples and the Primitive Roots Modulo a Prime

In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let p be an odd prime large enough. Then, there must exist three primitive roots x , y , and z modulo p such that x 2 + y 2 = z 2 .

Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles

The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more “Metallic” than the Pentagons, Octagons or any other (n2+4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means.

A SHORT NOTE ON FIBONACCI NUMBERS IN A GENERALIZED PYTHAGOREAN TRIPLES

This paper aims to construct a new formula that generates a Fibonacci numbers in a generalized Pythagorean triples. In addition, the paper formulates some Fibonacci identities and discuss some important findings.